Solving "Number of Beautiful Pairs" with a Twist: Efficient Coprime Calculation 

Introduction

In this article, we'll tackle the LeetCode exercise "Number of Beautiful Pairs" and explore an efficient solution that leverages a clever trick to calculate coprime numbers. We'll also discuss why implementing the standard GCD algorithm might not be necessary in this case.

Problem Statement

Given a 0-indexed integer array nums, find the total number of beautiful pairs, where a pair of indices i, j is called beautiful if the first digit of nums[i] and the last digit of nums[j] are coprime. Two integers x and y are coprime if there is no integer greater than 1 that divides both of them.

Solution

Here's a Python solution that uses a dictionary to store the coprime relationships between digits:

class Solution:

    def countBeautifulPairs(self, nums: List[int]) -> int:

        coprime_map = {}

        # ... (populate coprime map)

        num_beautiful_pairs = 0

        l = len(nums)

        for i in range(l):

            for j in range(i+1, l):

                digit_i = int(str(nums[i])[0])

                digit_j = nums[j] % 10

                if digit_j in coprime_map[digit_i]:

                    num_beautiful_pairs += 1

        return num_beautiful_pairs

The Coprime Map

Notice that we're not implementing the standard GCD algorithm to calculate the coprime relationships. Instead, we're using a precomputed dictionary coprime_map that stores the coprime digits for each digit from 1 to 9.

This is possible because we have a limited number of calculations to perform, and it's easy to precompute these relationships once. In fact, testing this solution against an array with all numbers from 1 to 9 was straightforward.

Tricks for Calculating First and Last Digits

In the solution above, we use two different tricks to calculate the first and last digits of a number:

• For the first digit, we simply convert the number to a string and take the first character: int(str(nums[i])[0]).

• For the last digit, we use the modulo operator to extract the last digit: nums[j] % 10.

These tricks are efficient and easy to understand, making them perfect for this problem.

Relevance to Data Engineering

This exercise has some relevance to data engineering in that it requires us to think about efficient ways to calculate relationships between numbers. In a real-world scenario, we might need to perform similar calculations on large datasets, where efficiency is crucial.

In particular, the use of precomputed dictionaries or lookup tables can be an effective way to speed up calculations, especially when dealing with limited domains like digits from 1 to 9.

Conclusion

Solving "Number of Beautiful Pairs" requires us to think creatively about coprime relationships and efficient calculation methods. By using a precomputed dictionary and clever tricks for calculating first and last digits, we can write an efficient solution that scales well. Additionally, this exercise highlights the importance of considering alternative approaches when solving problems in data engineering.

I hope you found this post helpful! Let me know if you have any questions or need further clarification on any of the points discussed.